Derive the Hopf fde for the characteristic functional (theta[y ; t]) for the pure IVP of the

Question:

Derive the Hopf fde for the characteristic functional $\theta[y ; t]$ for the pure IVP of the Burgers pde (1.2). Use the result obtained in Problem (9.1) to establish the solution operator and its inverse.

Problem 9.1

Solve the pure IVP for the Burgers pde (1.2) with initial condition $u(0, x)=u_{0}(x) \in L_{\mathcal{D}}^{2} \cap C_{\mathcal{D}}^{\infty}, \mathcal{D}=(-\infty, \infty)$, using the Hopf-Cole transformation (1.3).

Pde (1.2)